Mathematics
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Lazarus Fuchs
Lazarus Immanuel Fuchs was a Jewish-German mathematician who contributed important research in the field of linear differential equations. He was died in Berlin, Germany, he was buried in Schöneberg in the St. Matthew's Cemetery, his grave in section H is listed as a grave of honour of the State of Berlin. He is the eponym of Fuchsian groups and functions, the Picard–Fuchs equation. A singular point a of a linear differential equation y ″ + p y ′ + q y = 0 is called Fuchsian if p and q are meromorphic at the point a, have poles of orders at most 1 and 2, respectively. According to a theorem of Fuchs, this condition is necessary and sufficient for the regularity of the singular point, that is, to ensure the existence of two linearly independent solutions of the form y j = ∑ n = 0 ∞ a j, n n + σ j, a 0 ≠ 0 j = 1, 2. Where the exponents σ j can be determined from the equation. In the case when σ 1 − σ 2 is an integer this formula has to be modified. Another well-known result of Fuchs is the Fuchs's conditions, the necessary and sufficient conditions for the non-linear differential equation of the form F = 0 to be free of movable singularities.
Lazarus Fuchs was the father of a German mathematician. Über Funktionen zweier Variabeln, welche durch Umkehrung der Integrale zweier gegebener Funktionen entstehen, Göttingen 1881. Zur Theorie der linearen Differentialgleichungen, Berlin 1901. Gesammelte Werke, Hrsg. von Richard Fuchs und Ludwig Schlesinger. 3 Bde. Berlin 1904–1909. Media related to Lazarus Immanuel Fuchs at Wikimedia Commons Jeremy Gray. "Fuchs and the theory of differential equations". Bulletin of the AMS. New Series. 10: 1–26. Doi:10.1090/S0273-0979-1984-15186-3. MR 0722855. G. B. Mathews Lazarus Fuchs Nature 66:156,7. O'Connor, John J.. Lazarus Fuchs at the Mathematics Genealogy Project
Carl Wilhelm Borchardt
Carl Wilhelm Borchardt was a German mathematician. Borchardt was born to a Jewish family in Berlin, his father, was a respected merchant, his mother was Emma Heilborn. Borchardt studied including Julius Plücker and Jakob Steiner, he studied at the University of Berlin under Lejeune Dirichlet in 1836 and at the University of Königsberg in 1839. In 1848 he began teaching at the University of Berlin, he did research in the area of arithmetic-geometric mean, continuing work by Lagrange. He generalised the results of Kummer diagonalising symmetric matrices, using determinants and Sturm functions, he was an editor of Crelle's Journal from 1856–80, during which time it was known as Borchardt's Journal. He died in Germany, his grave is preserved in the Protestant Friedhof III der Jerusalems- und Neuen Kirchengemeinde in Berlin-Kreuzberg, south of Hallesches Tor. Cayley's formula
University of Königsberg
The University of Königsberg was the university of Königsberg in East Prussia. It was founded in 1544 as the world's second Protestant academy by Duke Albert of Prussia, was known as the Albertina. Following World War II, the city of Königsberg was transferred to the Soviet Union according to the 1945 Potsdam Agreement, renamed Kaliningrad in 1946; the Albertina was closed and the remaining German population expelled. Today, the Immanuel Kant Baltic Federal University in Kaliningrad claims to maintain the traditions of the Albertina. Albert, former Grand Master of the Teutonic Knights and first Duke of Prussia since 1525, had purchased a piece of land behind Königsberg Cathedral on the Kneiphof island of the Pregel River from the Samland chapter, where he had an academic gymnasium erected in 1542, he issued the deed of foundation of the Collegium Albertinum on 20 July 1544, after which the university was inaugurated on 17 August. The newly established Protestant duchy was a fiefdom of the Crown of the Kingdom of Poland and the university served as a Lutheran counterpart to the Catholic Cracow Academy.
Its first rector was son-in-law of Philipp Melanchthon. Lithuanian scholars Stanislovas Rapalionis and Abraomas Kulvietis were among the first professors of university. All professors had to take an oath on the Augsburg Confession. Since the Prussian lands lay beyond the borders of the Holy Roman Empire, both Emperor Charles V and Pope Paul III withheld their approval the Königsberg academy received the royal privilege by King Sigismund II Augustus of Poland on 28 March 1560. From 1618 the Prussian duchy was ruled in personal union by the Margraves of Brandenburg and in 1657 the "Great Elector" Frederick William of Brandenburg acquired full sovereignty over Prussia from Poland by the Treaty of Wehlau; the Albertina was the second oldest university and intellectual centre of Protestant Brandenburg-Prussia. It comprised four colleges: Theology, Medicine and Law also natural sciences. Subsequent rectors included numerous Hohenzollern Prussian royals, who had never been to the university represented by a prorector in charge of academic affairs.
The Prussian lands remained unharmed by the disastrous Thirty Years' War, which gained the Königsberg university an increasing popularity among students. In the 17th century, it was known as a home to Simon Dach, serving as rector in 1656/57, his fellow poets. Tsar Peter I of Russia visited the Albertina in 1697, leading to increased contacts between Prussia and the Russian Empire. Notable Russian students at Königbserg were Kirill Razumovsky president of the Russian Academy of Sciences and General Mikhail Andreyevich Miloradovich; the university and the city had profound impact on the development of Lithuanian culture. The first book in Lithuanian language was printed here in 1547 and several important Lithuanian writers attended the Albertina; the university was the preferred educational institution of the Baltic German nobility. The 18th century went down in cultural history as the "Königsberg Century" of Enlightenment, a heyday initiated by the Albertina student Johann Christoph Gottsched and continued by the philosopher Johann Georg Hamann and writer Theodor Gottlieb von Hippel the Elder.
Notable alumni were Johann Gottfried Herder, Zacharias Werner, Johann Friedrich Reichardt, E. T. A. Hoffmann, foremost the philosopher Immanuel Kant, rector in 1786 and 1788; these scholars laid the foundations for the Weimar Classicism and German Romanticism movements. The Albertina's magnificent botanical garden was inaugurated in 1811 during the Napoleonic Wars. Two years Friedrich Wilhelm Bessel established his outstanding observatory next door to the garden. Other university professors included such giants of the science world as the philosopher Johann Gottlieb Fichte, the biologist Karl Ernst von Baer, the mathematician Carl Gustav Jacobi, the mineralogist Franz Ernst Neumann and the physicist Hermann von Helmholtz. In the 19th and 20th centuries, the university was most famous for its school of Mathematics, founded by Carl Gustav Jacob Jacobi, continued by his pupils Ludwig Otto Hesse, Friedrich Richelot, Johann G. Rosenhain and Philipp Ludwig von Seidel, it was associated with the names of Hermann Minkowski, Adolf Hurwitz, Ferdinand von Lindemann and David Hilbert, one of the greatest modern mathematicians.
The mathematicians Alfred Clebsch and Carl Gottfried Neumann founded the Mathematische Annalen in 1868, which soon became the most influential mathematical journal of the time. Celebrating the university's 300 years jubilee 0n 31 August 1844, King Frederick William IV of Prussia laid the foundation for the new main building of the Albertina, inaugurated in 1862 by Crown Prince Frederick and Prorector Johann Karl Friedrich Rosenkranz; the building on central Paradeplatz was erected in a neo-Renaissance style according to plans designed by Friedrich August Stüler. The facade was adorned by an equestrian figure in relief of Albert of Prussia. Below it were niches containing statues of the Protestant reformers Martin Luther and Philipp Melanchthon. Inside was a handsome staircase, borne by marble columns; the Senate Hall contained a portrait of Emperor Frederick III by Lauchert and a bust of Immanuel Kant by Hagemann, a student of Schadow. The adjacent hall was adorned with frescoes painted in 1870.
The university library was situated on Mitteltragheim in 1901 and contained over 230,00
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors, he singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his successors. Cauchy was a prolific writer. Cauchy was the son of Louis François Marie-Madeleine Desestre. Cauchy had two brothers: Alexandre Laurent Cauchy, who became a president of a division of the court of appeal in 1847 and a judge of the court of cassation in 1849, Eugene François Cauchy, a publicist who wrote several mathematical works. Cauchy married Aloise de Bure in 1818, she was a close relative of the publisher. They had Marie Françoise Alicia and Marie Mathilde. Cauchy's father was a high official in the Parisian Police of the Ancien Régime, but lost this position due to the French Revolution, which broke out one month before Augustin-Louis was born.
The Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris. There Louis-François Cauchy found himself a new bureaucratic job in 1800, moved up the ranks; when Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, became Secretary-General of the Senate, working directly under Laplace. The famous mathematician Lagrange was a friend of the Cauchy family. On Lagrange's advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, the best secondary school of Paris at that time, in the fall of 1802. Most of the curriculum consisted of classical languages. In spite of these successes, Augustin-Louis chose an engineering career, prepared himself for the entrance examination to the École Polytechnique. In 1805, he placed second out of 293 applicants on this exam, he was admitted. One of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education.
The school functioned under military discipline, which caused the young and pious Cauchy some problems in adapting. He finished the Polytechnique in 1807, at the age of 18, went on to the École des Ponts et Chaussées, he graduated with the highest honors. After finishing school in 1810, Cauchy accepted a job as a junior engineer in Cherbourg, where Napoleon intended to build a naval base. Here Augustin-Louis stayed for three years, was assigned the Ourcq Canal project and the Saint-Cloud Bridge project, worked at the Harbor of Cherbourg. Although he had an busy managerial job, he still found time to prepare three mathematical manuscripts, which he submitted to the Première Classe of the Institut de France. Cauchy's first two manuscripts were accepted. In September 1812, now 23 years old, Cauchy returned to Paris after becoming ill from overwork. Another reason for his return to the capital was that he was losing his interest in his engineering job, being more and more attracted to the abstract beauty of mathematics.
Therefore, when his health improved in 1813, Cauchy chose to not return to Cherbourg. Although he formally kept his engineering position, he was transferred from the payroll of the Ministry of the Marine to the Ministry of the Interior; the next three years Augustin-Louis was on unpaid sick leave, spent his time quite fruitfully, working on mathematics. He attempted admission to the First Class of the Institut de France but failed on three different occasions between 1813 and 1815. In 1815 Napoleon was defeated at Waterloo, the newly installed Bourbon king Louis XVIII took the restoration in hand; the Académie des Sciences was re-established in March 1816. The reaction of Cauchy's peers was harsh. In November 1815, Louis Poinsot, an associate professor at the École Polytechnique, asked to be exempted from his teaching duties for health reasons. Cauchy was by a rising mathematical star, who merited a professorship. One of his great successes at that time was the proof of Fermat's polygonal number theorem.
However, the fact that Cauchy was known to be loyal to the Bourbon
Braniewo
Braniewo, is a town in northeastern Poland, in the Warmian-Masurian Voivodeship, with a population of 18,068. It is the capital of Braniewo County. Braniewo lies on the Pasłęka River about 5 km from the Vistula Lagoon, about 35 km northeast of Elbląg and 55 km southwest of Kaliningrad; the Polish border with Russia's Kaliningrad Oblast lies 6 km north, may be reached from Braniewo via National Highway 54. According to the German geographer Johann Friedrich Goldbeck, the town was named Brunsberg after Bruno von Schauenburg, bishop of Olmütz in Moravia, who accompanied King Ottokar II of Bohemia in 1254 and 1267 when the latter participated in the crusade of the Teutonic Knights against the Old Prussians, it has been suggested that the name Braunsberg might stem from Brusebergue, but this notion is not documented. In 1243 the settlement and the surrounding region of Warmia was given by the Teutonic Order to the newly created Diocese of Ermland, whose bishop built his cathedral in the town and made it his chief residence.
The city was granted town privileges based on those of Lübeck in 1254, but in 1261 was destroyed and depopulated during the second of the Prussian Uprisings. It was settled by colonists from Lübeck. In 1284 it was given a new town charter, again based on that of Lübeck. However, the next bishop, Heinrich Fleming, transferred the chapter from Braunsberg to Frauenburg. In 1296 a Franciscan abbey was built, in 1342 a "new town" was added; as the most important trading and harbor city in Warmia, the town prospered as member of the Hanseatic League, which it remained until 1608. It remained a part of the monastic state of the Teutonic Knights until 1466, when as a consequence of the Second Peace of Thorn ending the Thirteen Years' War, it came under jurisdiction of the Kingdom of Poland as part of the new autonomous province of Royal Prussia. After the secularization of the Teutonic Order in 1525, a large part of its residents converted to Lutheran Protestantism. Duke Albert, grand master of the Order, sought to unite Warmia with Ducal Prussia, causing the Catholics of the town to swear allegiance to the king of Poland in return for aid against Protestant Prussia.
In 1526 a Polish royal commission released Braunsberg burghers from the oath to the Polish king and handed the town back to Prince-Bishop Mauritius Ferber. However, just like the entire area of Warmia, Braunsberg swore allegiance to the Prince-Bishops of Warmia, who were subjects of the popes. Additionally, it had to denounce all Lutheran teachings and hand over Lutheran writings. Thereafter Warmia, though inhabited in part by ethnic Germans, remained predominantly Roman Catholic. Braniewo was occupied by Sweden for about three years during the Livonian War in the 16th century. In Warmia, Lutheran teachings again were suppressed when Prince-Bishop Stanislaus Hosius brought in the Jesuits and founded the Collegium Hosianum school. A priestly seminary was added in 1564. Pope Gregory XIII added a papal mission seminary for northern and eastern European countries. Regina Protmann, a native of Braunsberg, founded the Saint Catherine Order of Sisters in the town, recognized by the church in 1583; the Jesuit theologian Antonius Possevinus was instrumental in enlarging the Collegium Hosianum in the 1580s to counter the growing Protestant movement.
The Polish, Catholic town was annexed by the Protestant Kingdom of Prussia in 1772 during the First Partition of Poland and made part of the province of East Prussia the following year. Braunsberg obtained its first railway connection with the rest of the kingdom via the Prussian Eastern Railway in 1852. In the early 20th century, the town was the leading academic center of East Prussia next to Königsberg. In 1912 the Jesuit college became the State Academy of Braunsberg. Prior to World War II, the population of Braunsberg had grown to more than 21,000, of whom 59 percent were listed as Catholic and 29 percent Protestant. In Braniewo and surroundings a powerful Germanization movement, directed from Berlin, took place, among others, prohibited native Poles from speaking Polish under penalty of imprisonment, or forbade those who declared Polish nationality from owning land; this action contributed to the artificially inflated numbers of "Germans" in the area. The Second World War turned much of the town into ruins.
After three and a half years of savage warfare, Soviet forces began their assault on German land by attacking East Prussia on Jan. 13, 1945. Red Army formations reached the Vistula Lagoon north of Braunsberg on Jan. 26. In early February German civilians began fleeing from Braunsberg across the ice of the frozen lagoon to the Vistula Spit, from which many journeyed to either Danzig Gdańsk, or Pillau and managed to board German ships that made the perilous voyage westward. Braunsberg was captured by Soviet troops on March 20, 1945. At the end of the war and thereafter, those German residents who had not fled or been killed were expelled to what remained of Germany. Under border changes promulgated at the Potsdam Conference, the region was returned to Poland and the town became Braniewo, being repopulated by Polish settlers, many from areas of eastern Poland annexed by the Soviet Union under terms of the 1939 Molotov–Ribbentrop Pact. Heavy fighting and wanton
West Prussia
The Province of West Prussia was a province of Prussia from 1773 to 1829 and 1878 to 1922. West Prussia was established as a province of the Kingdom of Prussia in 1773, formed from Royal Prussia of the Polish-Lithuanian Commonwealth annexed in the First Partition of Poland. West Prussia was dissolved in 1829 and merged with East Prussia to form the Province of Prussia, but was re-established in 1878 when the merger was reversed and became part of the German Empire. From 1918, West Prussia was a province of the Free State of Prussia within Weimar Germany, losing most of its territory to the Second Polish Republic and the Free City of Danzig in the Treaty of Versailles. West Prussia was dissolved in 1922, its remaining western territory was merged with Posen to form Posen-West Prussia, its eastern territory merged with East Prussia as the Region of West Prussia district. West Prussia's provincial capital alternated between Danzig during its existence. West Prussia was notable for its ethnic and religious diversity due to immigration and cultural changes, with the population becoming mixed over the centuries.
Since the early Middle Ages the region was inhabited by numerous Slavic and Baltic peoples, such as Pomeranians in the Pomerelia region, Old Prussians and Masovians in Kulmerland, Pomesanians east of the Vistula River. Germans followed. Germans were the largest group in West Prussia until its dissolution in 1922, with large numbers of Kashubians, Poles and Jews settling in the region. In the Thirteen Years' War, the towns of the Prussian Confederation in Pomerelia and the adjacent Prussian region east of the Vistula River rebelled against the rule of the Teutonic Knights and sought the assistance of King Casimir IV Jagiellon of Poland. By the Second Peace of Thorn in 1466, Pomerelia and the Prussian Culm and Marienburg lands as well as the autonomous Prince-Bishopric of Warmia became the Polish province of Royal Prussia, which received special rights in Danzig; the province became a Land of the Polish Crown within the Polish–Lithuanian Commonwealth by the 1569 Union of Lublin. East Prussia around Königsberg, on the other hand, remained with the State of the Teutonic Knights, who were reduced to vassals of the Polish kings.
Their territory was secularised to the Duchy of Prussia according to the 1525 Treaty of Kraków. Ruled in personal union with the Imperial Margraviate of Brandenburg from 1618, the Hohenzollern rulers of Brandenburg-Prussia were able to remove the Polish suzerainty by the 1657 Treaty of Wehlau; this development turned out to be fatal to the Polish monarchy, as the two parts of the rising Kingdom of Prussia were separated by Polish land. In the 1772 First Partition of Poland the Prussian king Frederick the Great took the occasion to annex most of Royal Prussia; the addition gave Prussia a land connection between the Province of Pomerania and East Prussia, cutting off the Polish access to the Baltic Sea and rendering East Prussia more defensible in the event of war with the Russian Empire. The annexed voivodeships of Pomerania except for the City of Danzig and Kulm were incorporated into the Province of West Prussia the following year, while Ermland became part of the Province of East Prussia.
Further annexed areas of Greater Poland and Kuyavia in the south formed the Netze District. The Partition Sejm ratified the cession on 30 September 1773. Thereafter Frederick styled himself "King of Prussia" rather than "King in Prussia." The Polish administrative and legal code was replaced by the Prussian system, 750 schools were built from 1772-1775. Both Protestant and Roman Catholic teachers taught in West Prussia, teachers and administrators were encouraged to be able to speak both German and Polish. Frederick II of Prussia advised his successors to learn Polish, a policy followed by the Hohenzollern dynasty until Frederick III decided not to let William II learn Polish. Despite this, Frederick II looked askance upon many of his new citizens. In a letter from 1735, he calls them "dirty" and "vile apes" He had nothing but contempt for the szlachta, the numerous Polish nobility, wrote that Poland had "the worst government in Europe with the exception of Ottoman Empire", he considered West Prussia less civilized than Colonial Canada and compared the Poles to the Iroquois.
In a letter to his brother Henry, Frederick wrote about the province that "it is a good and advantageous acquisition, both from a financial and a political point of view. In order to excite less jealousy I tell everyone that on my travels I have seen just sand, pine trees, heath land and Jews. Despite that there is a lot of work to be done. Frederick invited German immigrants to redevelop the province. Many German officials regarded the Poles with contempt. According to the Polish historian Jerzy Surdykowski, Frederick the Great introduced 300,000 German colonists. According to Christopher Clark, 54 percent of the annexed area's and 75 percent of the urban population were German-speaking Protestants. Further Polish areas were annexed in the Second Partition of Poland in 1793, now including the cities of Danzig and Thorn; some of the areas of Greater Poland annexed in 1772 that formed the Netze District were added to West Prussia in 1793 as well. After the defeat of Prussia by the Napoleonic French Empire at the 1806 Battle of Jena-Auerstedt, West Prussia lost its southern territory